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Properties

Label 2-3e3-9.2-c4-0-1
Degree $2$
Conductor $27$
Sign $0.542 - 0.839i$
Analytic cond. $2.79098$
Root an. cond. $1.67062$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.67 + 3.27i)2-s + (13.4 + 23.3i)4-s + (−10.2 + 5.89i)5-s + (26.6 − 46.1i)7-s + 71.6i·8-s − 77.2·10-s + (−108. − 62.4i)11-s + (37.3 + 64.6i)13-s + (302. − 174. i)14-s + (−19.2 + 33.3i)16-s − 7.70i·17-s − 54.1·19-s + (−274. − 158. i)20-s + (−409. − 709. i)22-s + (−346. + 199. i)23-s + ⋯
L(s)  = 1  + (1.41 + 0.819i)2-s + (0.841 + 1.45i)4-s + (−0.408 + 0.235i)5-s + (0.543 − 0.941i)7-s + 1.11i·8-s − 0.772·10-s + (−0.894 − 0.516i)11-s + (0.220 + 0.382i)13-s + (1.54 − 0.890i)14-s + (−0.0752 + 0.130i)16-s − 0.0266i·17-s − 0.149·19-s + (−0.687 − 0.396i)20-s + (−0.845 − 1.46i)22-s + (−0.654 + 0.378i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.542 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $0.542 - 0.839i$
Analytic conductor: \(2.79098\)
Root analytic conductor: \(1.67062\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{27} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 27,\ (\ :2),\ 0.542 - 0.839i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.20507 + 1.20070i\)
\(L(\frac12)\) \(\approx\) \(2.20507 + 1.20070i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-5.67 - 3.27i)T + (8 + 13.8i)T^{2} \)
5 \( 1 + (10.2 - 5.89i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (-26.6 + 46.1i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (108. + 62.4i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-37.3 - 64.6i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 7.70iT - 8.35e4T^{2} \)
19 \( 1 + 54.1T + 1.30e5T^{2} \)
23 \( 1 + (346. - 199. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-468. - 270. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-766. - 1.32e3i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 1.71e3T + 1.87e6T^{2} \)
41 \( 1 + (1.09e3 - 629. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.30e3 + 2.25e3i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-692. - 400. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 4.22e3iT - 7.89e6T^{2} \)
59 \( 1 + (-2.88e3 + 1.66e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-7.50 + 13.0i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (2.59e3 + 4.48e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 1.92e3iT - 2.54e7T^{2} \)
73 \( 1 - 949.T + 2.83e7T^{2} \)
79 \( 1 + (-118. + 206. i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-1.14e4 - 6.58e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 575. iT - 6.27e7T^{2} \)
97 \( 1 + (7.56e3 - 1.30e4i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.27517608180815432630011683150, −15.44225922513210945833368648650, −14.13948176740822078101491159280, −13.50148043035023279377233560335, −12.02855121276468878187464917440, −10.64011568617985398607858488925, −8.032665557667094727391794166392, −6.84359344130664145247730523403, −5.12925266861374119419030227904, −3.64310623844618054977556306516, 2.43572066286306344558225358102, 4.44183448109127015879252124546, 5.76035861843139599346352821905, 8.207030048065230536764804228261, 10.34964006019452818590092214952, 11.72685869894726959868834067016, 12.46810537363915096366668105345, 13.67074326834768700813595069632, 14.99512396493155284717853122207, 15.73955536854205640225580783122

Graph of the $Z$-function along the critical line